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In this paper, we present an entropy-stable (ES) discretization using a nodal discontinuous Galerkin (DG) method for the ideal multi-ion magneto-hydrodynamics (MHD) equations. We start by performing a continuous entropy analysis of the ideal multi-ion MHD system, described by, e.g., Toth (2010) [Multi-Ion Magnetohydrodynamics], which describes the motion of multi-ion plasmas with independent momentum and energy equations for each ion species. Following the continuous entropy analysis, we propose an algebraic manipulation to the multi-ion MHD system, such that entropy consistency can be transferred from the continuous analysis to its discrete approximation. Moreover, we augment the system of equations with a generalized Lagrange multiplier (GLM) technique to have an additional cleaning mechanism of the magnetic field divergence error. We first derive robust entropy-conservative (EC) fluxes for the alternative formulation of the multi-ion GLM-MHD system that satisfy a Tadmor-type condition and are consistent with existing EC fluxes for single-fluid GLM-MHD equations. Using these numerical two-point fluxes, we construct high-order EC and ES DG discretizations of the ideal multi-ion MHD system using collocated Legendre--Gauss--Lobatto summation-by-parts (SBP) operators. The resulting nodal DG schemes satisfy the second-law of thermodynamics at the semi-discrete level, while maintaining high-order convergence and local node-wise conservation properties. We demonstrate the high-order convergence, and the EC and ES properties of our scheme with numerical validation experiments. Moreover, we demonstrate the importance of the GLM divergence technique and the ES discretization to improve the robustness properties of a DG discretization of the multi-ion MHD system by solving a challenging magnetized Kelvin-Helmholtz instability problem that exhibits MHD turbulence.